To simplify the given expression [tex]\(\frac{5^x \cdot 5^{x-1}}{4-5^{x-1}}\)[/tex], let's break down the steps systematically:
1. Simplify the Numerator:
The given numerator is [tex]\(5^x \cdot 5^{x-1}\)[/tex].
Recall the properties of exponents: [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]
Therefore:
[tex]\[
5^x \cdot 5^{x-1} = 5^{x + (x-1)} = 5^{2x-1}
\][/tex]
So, the numerator simplifies to [tex]\(5^{2x-1}\)[/tex].
2. Write Down the Denominator:
The denominator is given as [tex]\(4 - 5^{x-1}\)[/tex]. This part of the expression remains as is, as it is already in its simplest form.
3. Combine the Simplified Forms:
Now, we combine the simplified numerator and the given denominator to rewrite the expression in its simplified form.
Thus, the simplified form of the given expression becomes:
[tex]\[
\frac{5^{2x-1}}{4 - 5^{x-1}}
\][/tex]
Putting it all together, the fully simplified expression is:
[tex]\[
\boxed{\frac{5^{2x-1}}{4 - 5^{x-1}}}
\][/tex]
